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Chapter 3: Problem 55

Concept Check In Exercises \(55-62,\) describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$3 x=y-9$$

Short Answer

Expert verified
The graph is a straight line with a slope of 3 and a y-intercept of 9.

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation in the standard form of a linear equation. The given equation is \(3x = y - 9\). To make it easier to understand, solve for \(y\). Add 9 to both sides: \[3x + 9 = y\] This can be rewritten as \[y = 3x + 9\]
02

Identify the Slope and Intercept

In the equation \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m\) is 3, and the y-intercept \(b\) is 9. This tells that the line rises 3 units for every 1 unit it moves to the right, and it crosses the y-axis at 9.
03

Sketch the Graph

First, plot the y-intercept (0, 9) on the coordinate plane. Then, use the slope to find another point on the line. From (0, 9), move up 3 units and to the right 1 unit to get (1, 12). Draw a line passing through these points. This line represents the equation \(y = 3x + 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Linear Equations
Graphing linear equations involves plotting points on the coordinate plane and drawing a line through those points. This process helps us visualize the equation and how it behaves. To graph any linear equation, follow these general steps:

  • Rewrite the equation in slope-intercept form, if needed.
  • Identify the slope and y-intercept from the equation.
  • Plot the y-intercept on the coordinate plane.
  • Use the slope to find another point on the line.
  • Draw a line through these points to complete the graph.

For instance, if we have the equation \(y = 3x + 9\), you would start by plotting the y-intercept (0, 9). Then using the slope, which is 3, move up 3 units and to the right 1 unit from the y-intercept to find another point, (1, 12). Connect these points with a straight line.
Slope-Intercept Form
Slope-intercept form is a way of writing linear equations that makes it easy to understand the slope and y-intercept of the line. The general format is \ y = mx + b \. Here, \(m\) represents the slope, and \(b\) represents the y-intercept.

The slope tells you how steep the line is and the direction it goes. It is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. For example, a slope of 3 means you move up 3 units for every 1 unit you move to the right. The y-intercept, on the other hand, is the point where the line crosses the y-axis. This happens when \ x = 0 \. In our example, \ y = 3x + 9 \, the y-intercept is 9.

Rewriting an equation into slope-intercept form can make graphing and understanding the line much easier. For the given problem, converting \(3x = y - 9\) into slope-intercept form gives us \( y = 3x + 9 \).
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph equations. It has two axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin (0, 0). Each point on the plane is identified by an ordered pair \ (x, y) \, where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.

When graphing a linear equation, plot points on this plane based on the slope and y-intercept. For example, with \ y = 3x + 9 \, you start by plotting the y-intercept (0, 9). Then, using the slope, you can find more points. From (0, 9), move up 3 units (rise) and 1 unit to the right (run) to plot the next point (1, 12). Connect these points with a straight line, and you have the graph of the equation.

Understanding how to use the coordinate plane is essential for graphing not just linear equations but any mathematical function.

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Most popular questions from this chapter

The table shows the rate (in percent) at which 2-year college students (public) completed a degree within 3 years. $$ \begin{array}{|c|c|} \hline \text { Year } & {\text { Percent }} \\ \hline 2002 & {31.6} \\ {2003} & {30.1} \\ {2004} & {29.0} \\ {2005} & {27.5} \\ {2006} & {26.6} \\ {2007} & {26.9} \\ \hline \end{array} $$ (a) Write the data from the table as ordered pairs \((x, y),\) where \(x\) represents the year and \(y\) represents the percent. (b) What does the ordered pair \((2007,26.9)\) mean in the context of this problem? (c) Make a scatter diagram of the data, using the ordered pairs from part (a) and the given grid. (d) Describe the pattern indicated by the points on the scatter diagram. What is happening to rates at which 2 -year college students complete a degree within 3 years?

Suppose that it costs 5000 dollars to start up a business selling snow cones. Furthermore, it costs 0.50 dollars per cone in labor, ice, syrup, and overhead. Then the cost to make \(x\) snow cones is given by \(y\) dollars, where $$ y=0.50 x+5000 $$ Express each of the following as an ordered pair. (a) When 100 snow cones are made, the cost is 5050 dollars. (b) When the cost is 6000 dollars, the number of snow cones made is 2000 .

Graph each line passing through the given point and having the given slope. $$ (-2,3), m=0 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines in Section \(3.3 .)\) Through \((2,3) ; \quad\) parallel to \(4 x-y=-2\)

Graph each equation by using the slope and y-intercept. $$ y=3 x+2 $$
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